Deduction Systems for Coalgebras Over Measurable Spaces

Abstract

A theory of infinitary deduction systems is developed for the modal logic of coalgebras for measurable polynomial functors on the category of measurable spaces. These functors have been shown by Moss and Viglizzo to have final coalgebras that represent certain universal type spaces in game-theoretic economics. A notable feature of the deductive machinery is an infinitary Countable Additivity Rule.A deductive construction of canonical spaces and coalgebras leads to completeness results. These give a proof-theoretic characterisation of the semantic consequence relation for the logic of any measurable polynomial functor as the least deduction system satisfying Lindenbaum's Lemma. It is also the only Lindenbaum system that is sound.The theory is additionally worked out for Kripke polynomial functors, on the category of sets, that have infinite constant sets in their formation.

Strong Completeness for Markovian Logics

“…We are confident that similar results can be obtained for the gen-eral case of the measurable polynomial functors on the category of measurable spaces considered in [11], but we do not have such a result yet.…”

“…In related papers, to prove a similar result a so-called countable additivity axiom was used. This is an infinitary axiom with uncountable instances [11,21,15].…”

“…It is therefore tempting to understand this logic from a proof theoretic perspective. Recent papers [4,11,19] have established complete proof systems and prove finite model properties for similar logics. Goldblatt in [11] presents a proof-theoretic analysis of the logic of T -coalgebras, where T is any polynomial functor constructed from a standard monad on the category of measurable spaces.…”

“…Recent papers [4,11,19] have established complete proof systems and prove finite model properties for similar logics. Goldblatt in [11] presents a proof-theoretic analysis of the logic of T -coalgebras, where T is any polynomial functor constructed from a standard monad on the category of measurable spaces. He proves that the semantic consequence relation over T -coalgebras is equal to the least deducibility relation that satisfies Lindenbaum's lemma, which states that any consistent set of formulas can be extended to a maximally consistent set.…”

“…In other words, this consequence relation is equal to the least of all deducibility relations if and only if that least deducibility relation satisfies Lindenbaum's lemma. These logics are not compact and for proving the aforementioned results in [11] it is used a powerful infinitary axiom scheme named the Countable Additivity Rule (CAR). In [21] Zhou and Ying prove that such a logic is not strongly-complete in the absence of CAR.…”